\documentclass[twoside]{article}
\usepackage{amssymb} % used for R in Real numbers
\pagestyle{myheadings}
\markboth{\hfil Boundary-value problems \hfil EJDE--1999/09}
{EJDE--1999/09\hfil Idris Addou \& Abdelhamid Benmeza\"\i \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 1999}(1999), No.~09, pp. 1--29. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
\vspace{\bigskipamount} \\
%
Boundary-value problems for the one-dimensional
p-Laplacian with even superlinearity
\thanks{ {\em 1991 Mathematics Subject Classifications:} 34B15, 34C10.
\hfil\break\indent
{\em Key words and phrases:} One-dimensional p-Laplacian, two-point
boundary-value problem, \hfil\break\indent
superlinear, time mapping. \hfil\break\indent
\copyright 1999 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted October 28, 1998. Published March 8, 1999.} }
\date{}
%
\author{Idris Addou \& Abdelhamid Benmeza\"\i}
\maketitle
\begin{abstract}
This paper is concerned with a study of the quasilinear problem
$$ \displaylines{
-(|u'|^{p-2}u')'= |u|^p-\lambda ,\quad\mbox{in } (0,1)\,, \cr
u(0) =u(1) =0\,, \cr}
$$
where $p>1$ and $\lambda \in {\mathbb R}$ are parameters.
For $\lambda >0$, we determine a lower bound for the number of solutions
and establish their nodal properties.
For $\lambda \leq 0$, we determine the exact number of solutions.
In both cases we use a quadrature method.
\end{abstract}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\section{Introduction} \label{sec1}
This paper is devoted to a study of existence and multiplicity of solutions
to the quasilinear two-point boundary-value problem
\begin{eqnarray}
&-(\varphi _p(u') )'= f(\lambda,u) ,\quad\mbox{in }(0,1)\,,& \label{AD} \\
&u(0) = u(1)=0\,, & \nonumber
\end{eqnarray}
where $\varphi _p(s) =| s|^{p-2}s$ and $f(\lambda,u)= |u|^p-\lambda$.
Here $(\varphi _p(u') )'$ is the one-dimensional $p$-Laplacian, and $p>1$.
When the differential operator is linear, i.e., $p=2$, several existence and
multiplicity results, related to superlinear boundary value problems with
Dirichlet boundary data, are available in the literature. Let us recall some
of them for the one-dimensional case.
Lupo et al \cite{Lupo} have studied the non-autonomous case
\begin{eqnarray}
&-u''(x) =u^2(x) -t\sin x ,\quad\mbox{in }(0,\pi )\,,& \label{lupo} \\
&u(0) =u(\pi ) =0\,.& \nonumber
\end{eqnarray}
Using a combination of shooting and topological arguments, they show that
for any $k\in {\mathbb N}$ there exists $t_k>0$ such that for all
$t\geq t_k$, problem (\ref{lupo}) admits at least $k$ solutions.
Castro and Shivaji \cite{Castro and Shivaji}, using phase-plane analysis,
consider the problem
\begin{eqnarray}
&-u''(x) =g(u(x)) -\rho (x) -t ,\quad\mbox{in }(0,1)\,, &\label{Castro}\\
&u(0) =u(1) =0\,,& \nonumber
\end{eqnarray}
where $\rho $ a continuous function on $[0,1]$, $g\in C^1({\mathbb R})$,
$$
\lim _{s\rightarrow -\infty }\frac{g(s) }s=M\in
{\mathbb R},\quad\mbox{and}\quad \lim _{s\rightarrow +\infty }
\frac{g(s) }{s^{1+\sigma }}=+\infty \quad\mbox{with}\quad \sigma >0\,.
$$
They show that for $k\in {\mathbb N}$ there exists $t_k(M) $ such that
$\lim_{k\rightarrow +\infty} t_k(M) =+\infty $, and for all $t>t_k$,
problem (\ref{Castro}) has
at least two solutions with $k$ nodes in $(0,1)$.
The autonomous case has been studied by many authors. Let us mention some of
them. Independently of Castro and Shivaji, Ruf and Solimini
\cite{RufSolimini} consider the problem
\begin{eqnarray}
&-u''(x) =g(u(x)) -t ,\quad\mbox{in }(0,\pi )\,, &\label{Ruf} \\
&u(0)=u(\pi ) =0\,,& \nonumber
\end{eqnarray}
where
$$
g\in C^1({\mathbb R}) ,\quad \limsup_{s\rightarrow -\infty }g'(s) t_k$ problem (\ref
{Ruf}) has at least $k$ distinct solutions.
Prior to the papers mentioned above, Scovel \cite{Scovel} obtained the same
result as Ruf and Solimini \cite{RufSolimini} in the special case where
$g(u) =6u^2$. He has shown that for any integer $k\geq 1$, there
exist values $t_1t_k$ problem (\ref{Ruf})
(with $g(u) =6u^2$) admits at least $k$ distinct solutions.
Independently and prior to Scovel, in 1983, Ammar Khodja \cite{AmmarKhodja}
obtained a complete description of the solution set of the problem
\begin{eqnarray}
&-u''(x) =u^2(x) -\lambda ,\quad\mbox{in }(0,1)\,, &\label{AK} \\
&u(0) =u(1) =0\,. & \nonumber
\end{eqnarray}
He detects all the solutions to (\ref{AK}) for any value of $\lambda \in
{\mathbb R}$, and thus obtains the exact number of solutions to
(\ref {AK}) for all $\lambda$. To state his result,
for any integer $k\geq 1$, denote
$$
\displaylines {
S_k^+=\left\{ u\in C_0^2[0,1] :u'(0)
>0, u
\mbox{ admits }k-1\mbox{ nodes in }(0,1) \right\}, \cr
S_k^-=-S_k^+\quad \mbox{and}\quad S_k=S_k^+\cup S_k^-. \cr}
$$
\begin{theorem}
\cite{AmmarKhodja} There exists a sequence $(\lambda _k) _{k\geq
0}$ such that
$$
-\infty \lambda _1$, there is no positive solution.
\item[(ii)] If $\lambda >0$, there exists a unique solution in $S_1^-$.
\item[(iii)] If (and only if ) $\lambda >\lambda _k:$
\begin{itemize}
\item there exist exactly 2 solutions in $S_{2k}$
\item there exists exactly one solution in $S_{2k+1}^-$
\end{itemize}
\item[(iv)] There exists a sequence $(\mu _k) _{k\geq 1}$ such that
$$
\lambda _1\lambda _{k+1}$, there
exists a unique solution in $S_{2k+1}^+$.
\end{description}
\end{theorem}
The objective of this paper is to extend Ammar Khodja's result
to the general quasilinear case $p>1$. In particular, we will
show that if $\lambda \leq 0$ the same result holds for all $p>1$, but if
$\lambda >0$ and $p>2$ the situation is different from that obtained in
\cite{AmmarKhodja}. So, the behavior of the solution set of problem (\ref{AD})
depends not only on the values of $\lambda $ (as was shown in \cite
{AmmarKhodja}) but also on those of the parameter $p$.
These changes in the behavior of the solution set when the parameter $p$
varies is not new in the literature. Guedda and Veron \cite{Guedda-Veron}
consider the problem
\begin{eqnarray}
&-(\varphi _p(u') )' = \lambda \varphi _p(u) -f(u) ,\quad\mbox{in }
(0,1)\,,&
\label{GV} \\
& u(0) = u(1) = 0\,, &\nonumber
\end{eqnarray}
where $f$ is a $C^1$ odd function such that the function $s\mapsto f(
s) /s^{p-1}$ is strictly increasing on $(0,+\infty ) $
with limit $0$ at $0$ and $\lim _{s\rightarrow +\infty }f(s)
/s^{p-1}=+\infty $. They denote by $E_\lambda $ the solution set of problem
(\ref{GV}) and show, under some technical assumptions, that when $1

2$. \medskip
This paper is organized as follows. In Section \ref{sec2} we introduce
notation and state the main results (Theorems \ref{thm1} and
\ref{thm2}). Section \ref{sec3} is devoted to explain of the method
used in proving our results. In Section \ref{sec4} we prove Theorem
\ref{thm1}
and finally, in Section \ref{sec5}, we prove Theorem \ref{thm2}.
\section{ Notation and main results} \label{sec2}
In order to state the main
results, for any $k\in {\mathbb N}^*$, let
$$
S_k^+=\left\{ \begin{array}{r}
u\in C^1([\alpha ,\beta ] ) : \mbox{ $u$ admits exactly $(k-1)$
zeros in $(\alpha,\beta )$}\\ \mbox{ all simple, $u(\alpha ) =u(\beta) =0$
and
$u'(\alpha ) >0$} \end{array}
\right\}\,,
$$
$S_k^-=-S_k^+$ and $S_k=S_k^+\cup S_k^-$.
\paragraph{Definition}
Let $u\in C([\alpha ,\beta ] ) $ be a function with two consecutive zeros
$x_11$, $u$ is $((\beta -\alpha ) /k) -$
periodic.
\item Every hump of $u$ (necessarily positive) is symmetrical about the
center of the interval of its definition.
\item The derivative of each hump of $u$ vanishes once and only once.
\end{itemize}
Let $B_k^-=-B_k^+$ and $B_k=B_k^+\cup B_k^-$.
The first result concerns the case $\lambda \leq 0$ and gives the
{\em exact} number of solutions to (\ref{AD}).
\begin{theorem}[Case $\lambda \leq 0$]
\label{thm1} There exists a number $\lambda _{*}<0$ such that:
\begin{description}
\item[(i)] If $\lambda 0$.
\begin{theorem}[Case $\lambda >0$]
\label{thm2} For any $p>1$ there exist two real numbers $J(p)
>J_+(p) >0$ and for all $p>2$ there exists a positive real
number $J_{-}(p) 0$) admits a solution in $
A_1^+$ if and only if $00$) {\bf or} ($p>2$ and $0(2nJ(p)
) ^{p^2}$ {\bf or} $p>2$ and
\begin{eqnarray*}
\lefteqn{ \inf \left\{ (2nJ(p) ) ^{p^2},(2n(
J_{-}(p) +J_+(p) ) ) ^{p^2}\right\} }\\
(2(n+1) J(p)
) ^{p^2}$ {\bf or} $p>2$ and
\begin{eqnarray*}
\lefteqn{ \inf \left\{ (2(n+1) J(p) )
^{p^2},(2((n+1) J_+(p) +nJ_{-}(p) ) ) ^{p^2}\right\} }\\
(2nJ(p) ) ^{p^2}$
{\bf or} $p>2$ and
\begin{eqnarray*}
\lefteqn{ \inf \left\{ (2nJ(p) ) ^{p^2},(2(
(n+1) J_{-}(p) +nJ_+(p) )) ^{p^2}\right\} }\\
0$ and $p\in
(1,2] $ then\\ $\tilde S\subset (\bigcup\limits_{k\geq
1}A_k) \cup (\bigcup\limits_{k\geq 1}B_k) $, where
$\tilde S$ denotes the solution set of problem (\ref{AD}).
\paragraph{Remark}
The results obtained in \cite{AmmarKhodja}, for $p=2$, concerning solutions
in $A_{2n}$, $A_{2n+1}^-$, and $A_{2n+1}^+$, are more precise than those
stated in Theorem \ref{thm2}, assertions {\bf (v), (vi) }and {\bf (vii)} for
$p\neq 2$. In fact, these assertions do not provide the exact number of
solutions in $A_{2n}$, $A_{2n+1}^-$, and $A_{2n+1}^+$. The proof given
in \cite{AmmarKhodja} uses strongly the fact that the nonlinearity $u\mapsto
u^2-\lambda $ is a second degree polynomial function. We were not able to
obtain the same degree of precision.
\section{The method used }\label{sec3}
To obtain our results, we
make use of the well known time mapping approach. See, for instance,
Laetsch \cite{Laetsch}, de Mottoni \& Tesei \cite{Mottoni-Tesei1}, \cite
{Mottoni-Tesei2}, Smoller \& Wasserman \cite{Smoller-Wasserman}, Ammar
Khodja \cite{AmmarKhodja}, Shivaji \cite{Shivaji}, Guedda \& Veron \cite
{Guedda-Veron}, Ubilla \cite{Ubilla}, Man\'asevich et al \cite
{Mana-Njoku-Zano}, Addou \& Ammar Khodja \cite{AddouAmmarKhodja}, Addou et
al \cite{Addou-Boug-Derhab-Raffed}, Addou \& Benmeza\"I \cite{AddouBenmezai2}.
To describe this method we denote by $g$ a nonlinearity and by $p$ a real
parameter, and we assume one of the following conditions:
\begin{eqnarray}
& g\in C({\mathbb R},{\mathbb R}) \quad\mbox{and}\quad 1